Parallel solution of integral equation-based EM problems in the frequency domain

The future of computational electromagnetics is changing drastically as the new generation of computer chips evolves from single-core to multi-core. The burden now falls on software programmers to revamp existing codes and add new functionality to enable computational codes to run efficiently on this new generation of multi-core CPUs. In this book, you'll learn everything you need to know to deal with multi-core advances in chip design by employing highly efficient parallel electromagnetic code. Focusing only on the Method of Moments (MoM), the book covers:

In-Core and Out-of-Core LU Factorization for Solving a Matrix Equation

A Parallel MoM Code Using RWG Basis Functions and ScaLAPACK-Based In-Core and Out-of-Core Solvers A Parallel MoM Code Using Higher-Order Basis Functions and ScaLAPACK-Based In-Core and Out-of-Core Solvers Turning the Performance of a Parallel Integral Equation Solver Refinement of the Solution Using the Conjugate Gradient Method A Parallel MoM Code Using Higher-Order Basis Functions and Plapack-Based In-Core and Out-of-Core Solvers Applications of the Parallel Frequency Domain Integral Equation Solver

Appendices are provided with detailed information on the various computer platforms used for computation; a demo shows you how to compile ScaLAPACK and PLAPACK on the Windows® operating system; and a demo parallel source code is available to solve the 2D electromagnetic scattering problems.

Parallel Solution of Integral Equation-Based EM Problems in the Frequency Domain is indispensable reading for computational code designers, computational electromagnetics researchers, graduate students, and anyone working with CEM software.

Author Notes

Yu Zhang is an Associate Professor at Xidian University and currently works at Syracuse University. He authored the book Parallel Computation in Electromagnetics as well as over seventy journal papers and thirty conference papers. His research is focused on computational electromagnetics with an interest in antenna design, EMC simulation, and signal processing.
Tapan K. Sarkar is a Professor in the Department of Electrical and Computer Engineering at Syracuse University. His current research interests deal with numerical solutions of operator equations arising in electromagnetics and signal processing with applications in system design. He has authored or coauthored more than 300 journal articles, numerous conference papers, and thirty-two book chapters. He is the author of fifteen books, including Smart Antennas, History of Wireless, and Physics of Multiantenna Systems and Broadband Processing (all published by Wiley).

Preface	p. xiii
Acknowledgments	p. xvii
Acronyms	p. xix
Chapter 1 Introduction	p. 1
1.0 Summary	p. 1
1.1 A Brief Review of Parallel CEM	p. 1
1.1.1 Computational Electromagnetics	p. 1
1.1.2 Parallel Computation in Electromagnetics	p. 3
1.2 Computer Platforms Accessed in This Book	p. 9
1.3 Parallel Libraries Employed for the Computations	p. 12
1.3.1 ScaLAPACK - Scalable Linear Algebra Package	p. 13
1.3.2 PLAPACK - Parallel Linear Algebra Package	p. 16
1.4 Conclusion	p. 19
References	p. 19
Chapter 2 In-Core and Out-of-Core LU Factorization for Solving a Matrix Equation	p. 27
2.0 Summary	p. 27
2.1 Matrix Equation from a MoM Code	p. 28
2.2 An In-Core Matrix Equation Solver	p. 28
2.3 Parallel Implementation of an In-Core Solver	p. 32
2.3.1 Data Distribution for an LU Algorithm	p. 32
2.3.2 ScaLAPACK: Two-Dimensional Block-Cyclic Matrix Distribution	p. 36
2.3.3 PLAPACK: Physically Based Matrix Distribution	p. 38
2.3.4 Data Distribution Comparison between ScaLAPACK and PLAPACK	p. 40
2.4 Data Decomposition for an Out-of-Core Solver	p. 42
2.5 Out-of-Core LU Factorization	p. 43
2.5.1 I/O Analysis of Serial Right-Looking and Left-Looking Out-of-Core LU Algorithms	p. 45
2.5.1.1 Right-Looking Algorithm	p. 45
2.5.1.2 Left-Looking Algorithm	p. 47
2.5.2 Implementation of the Serial Left-Looking Out-of-Core LU Algorithm	p. 50
2.5.3 Design of a One-Slab Left-Looking Out-of-Core LU Algorithm	p. 55
2.6 Parallel Implementation of an Out-of-Core LU Algorithm	p. 61
2.6.1 Parallel Implementation of an Out-of-Core LU Algorithm Using ScaLAPACK	p. 61
2.6.2 Parallel Implementation of an Out-of-Core LU Algorithm Using PLAPACK	p. 64
2.6.3 Overlapping of the I/O with the Computation	p. 65
2.6.4 Checkpointing in an Out-of-Core Solver	p. 65
2.7 Solving a Matrix Equation Using the Out-of-Core LU Matrices	p. 66
2.8 Conclusion	p. 69
References	p. 69
Chapter 3 A Parallel MoM Code Using RWG Basis Functions and ScaLAPACK-Based In-Core and Out-of-Core Solvers	p. 71
3.0 Summary	p. 71
3.1 Electric Field Integral Equation (EFIE)	p. 71
3.2 Use of the Piecewise Triangular Patch (RWG) Basis Functions	p. 74
3.3 Testing Procedure	p. 76
3.4 Matrix Equation for MoM	p. 78
3.5 Calculation of the Various Integrals	p. 79
3.5.1 Evaluation of the Fundamental Integrals	p. 79
3.5.2 Extraction of the Singularity	p. 80
3.6 Calculation of the Fields	p. 81
3.7 Parallel Matrix Filling-In-Core Algorithm	p. 81
3.8 Parallel Matrix Filling-Out-of-Core Algorithm	p. 86
3.9 Numerical Results from a Parallel In-Core MoM Solver	p. 88
3.9.1 Numerical Results Compared with other Methods	p. 88
3.9.1.1 A PEC Cube	p. 88
3.9.1.2 A Combined Cube-and-Sphere PEC Model	p. 88
3.9.2 Different Metrics Used to Assess the Degree of Parallel Efficiency	p. 88
3.9.3 Efficiency and Portability of a Parallel MoM In-Core Solver	p. 92
3.10 Numerical Results from a Parallel Out-of-Core MoM Solver	p. 96
3.10.1 Parallel Out-of-Core Solver Can Be as Efficient as a Parallel In-Core Solver	p. 96
3.10.2 Scalability and Portability of the Parallel Out-of-Core Solver	p. 98
3.11 Conclusion	p. 104
References	p. 105
Chapter 4 A Parallel MoM Code Using Higher-Order Basis Functions and ScaLAPACK-Based In-Core and Out-of-Core Solvers	p. 107
4.0 Summary	p. 107
4.1 Formulation of the Integral Equation for Analysis of Dielectric Structures	p. 107
4.2 A General Formulation for the Analysis of Composite Metallic and Dielectric Structures	p. 110
4.3 Geometric Modeling of the Structures	p. 114
4.3.1 Right-Truncated Cone to Model Wire Structures	p. 114
4.3.2 Bilinear Surface for Modeling Arbitrarily Shaped Surfaces	p. 116
4.4 Higher-Order Basis Functions	p. 117
4.4.1 Current Expansion along a Thin PEC Wire	p. 117
4.4.2 Current Expansion over a Bilinear Surface	p. 119
4.5 Testing Procedure	p. 124
4.5.1 Testing Procedure for Thin PEC Wires	p. 124
4.5.2 Testing Procedure for Bilinear Surfaces	p. 119
4.6 Parallel In-Core and Out-of-Core Matrix Filling Schemes	p. 131
4.6.1 Parallel In-Core Matrix Filling Scheme	p. 132
4.6.2 Parallel Out-of-Core Matrix Filling Scheme	p. 134
4.7 Numerical Results Computed on Different Platforms	p. 136
4.7.1 Performance Analysis for the Parallel In-Core Integral Equation Solver	p. 136
4.7.1.1 Comparison of Numerical Results Obtained on Single-Core and Multicore Platforms	p. 136
4.7.1.2 Numerical Results Obtained on Single-Core Platforms	p. 141
4.7.1.2.1 Radiation from a Vivaldi Antenna Array	p. 141
4.7.1.2.2 Scattering from a Full-Size Airplane	p. 144
4.7.1.3 Numerical Results Obtained on Multicore Platforms	p. 146
4.7.2 Performance Analysis for the Parallel Out-of-Core Integral Equation Solver	p. 147
4.7.2.1 Vivaldi Antenna Array-a Large Problem Solved on Small Computer Platforms	p. 147
4.7.2.2 Solution for a Full-Size Airplane-Parallel Out-of-Core Solver Can Be as Efficient as the Parallel In-Core	p. 149
4.7.2.3 Solution for a Full-Size Airplane-Scalability and Portability of the Parallel Out-of-Core Solver	p. 150
4.7.2.4 Solution for a Full-Size Airplane-a Very Large Problem Solved on Nine Nodes of CEM-4	p. 153
4.8 Conclusion	p. 155
References	p. 155
Chapter 5 Tuning the Performance of a Parallel Integral Equation Solver	p. 157
5.0 Summary	p. 157
5.1 Anatomy of a Parallel Out-of-Core Integral Equation Solver	p. 157
5.1.1 Various Components of a Parallel Out-of-Core Solver that Can Be Observed through Ganglia and Tuned	p. 158
5.1.2 CPU Times of Parallel In-Core and Out-of-Core Integral Equation Solvers	p. 161
5.1.3 Performance of a Code Varies with the Amount of Storage Used on the Hard Disk	p. 165
5.2 Block Size	p. 170
5.3 Shape of the Process Grid	p. 173
5.4 Size of the In-Core Buffer Allocated to Each Process	p. 176
5.4.1 Optimizing IASIZE for a Parallel MoM Code Using Higher-Order Basis Functions	p. 177
5.4.1.1 Case A: Available 2 GB of RAM/Core	p. 177
5.4.1.1.1 Overview of Wall Time with Different IASIZE	p. 177
5.4.1.1.2 Details on Matrix Filling and Matrix Solving	p. 181
5.4.1.2 Case B: Available 4 GB of RAM/Core	p. 188
5.4.2 Optimizing IASIZE for a Parallel MoM Code Using RWG Basis Functions	p. 190
5.4.3 Influence of Physical RAM Size on Performance	p. 194
5.5 Relationship between Shape of the Process Grid and In-Core Buffer Size	p. 197
5.6 Overall Performance of a Parallel Out-of-Core Solver on HPC Clusters	p. 201
5.7 Conclusion	p. 205
References	p. 205
Chapter 6 Refinement of the Solution Using the Iterative Conjugate Gradient Method	p. 207
6.0 Summary	p. 207
6.1 Development of the Conjugate Gradient Method	p. 207
6.2 The Iterative Solution of a Matrix Equation	p. 212
6.3 Parallel Implementation of the CG Algorithm	p. 213
6.4 A Parallel Combined LU-CG Scheme to Refine the LU Solution	p. 215
6.5 Conclusion	p. 216
References	p. 217
Chapter 7 A Parallel MoM Code Using Higher-Order Basis Functions and PLAPACK-Based In-Core and Out-of-Core Solvers	p. 219
7.0 Summary	p. 219
7.1 Introduction	p. 219
7.2 Factors that Affect a Parallel In-Core and Out-of-Core Matrix Filling Algorithm	p. 220
7.3 Numerical Results	p. 224
7.3.1 Radiation from an Array of Vivaldi Antennas	p. 224
7.3.2 Scattering from an Electrically Large Aircraft	p. 228
7.3.3 Discussion of the Computational FLOPS Achieved	p. 230
7.4 Conclusion	p. 231
References	p. 231
Chapter 8 Applications of the Parallel Frequency-Domain Integral Equation Solver-TIDES	p. 233
8.0 Summary	p. 233
8.1 Performance Comparison between TIDES and a Commercial EM Analysis Software	p. 234
8.1.1 Analysis of a Scattering Problem	p. 234
8.1.2 Analysis of a Radiation Problem	p. 237
8.1.3 Analysis of a Coupling Problem	p. 241
8.2 EMC Prediction for Multiple Antennas Mounted on an Electrically Large Platform	p. 243
8.3 Analysis of Complex Composite Antenna Array	p. 248
8.4 Array Calibration for Direction-of-Arrival Estimation	p. 249
8.5 Radar Cross Section (RCS) Calculation of Complex Targets	p. 252
8.5.1 RCS Calculation of a Squadron of Tanks	p. 252
8.5.2 RCS of the Tanks inside a Forest Environment	p. 254
8.5.3 RCS from an Aircraft and a Formation of Aircraft	p. 257
8.5.4 RCS Simulation with Million Level Unknowns	p. 259
8.5.5 RCS of an Aircraft Carrier	p. 260
8.6 Analysis of Radiation Patterns of Antennas Operating Inside a Radome Along with the Platform on Which It Is Mounted	p. 264
8.7 Electromagnetic Interference (EMI) Analysis of a Communication System	p. 268
8.8 Comparison between Computations Using TIDES and Measurement data for Complex Composite Structures	p. 271
8.9 Conclusion	p. 273
References	p. 273
Appendix A A Summary of the Computer Platforms Used in This Book	p. 275
A.0 Summary	p. 275
A.1 Description of the Platforms Used in This Book	p. 275
A.2 Conclusion	p. 284
References	p. 285
Appendix B An Efficient Cross-Platform Compilation of the ScaLAPACK and PLAPACK Routines	p. 287
B.0 Summary	p. 287
B.1 Tools for Compiling both ScaLAPACK and PLAPACK	p. 287
B.2 Generating the ScaLAPACK Library	p. 288
B.2.1 Source Codes for Compiling ScaLAPACK	p. 288
B.2.2 Steps for Compiling ScaLAPACK	p. 288
B.2.3 Script Files for 32-bit Windows Operating System	p. 289
B.2.3.1 Script Files for BLAS	p. 289
B.2.3.2 Script Files for BLACS	p. 293
B.2.3.3 Script Files for ScaLAPACK	p. 296
B.2.4 Script Files for 64-bit Windows Operating System	p. 297
B.3 Generating the PLAPACK Library	p. 298
B.3.1 Source Codes for Compiling PLAPACK	p. 298
B.3.2 Script Files for PLAPACK	p. 298
B.4 Tuning the Performance by Turning on Proper Flags	p. 300
B.5 Conclusion	p. 301
References	p. 301
Appendix C An Example of a Parallel MoM Source Code for Analysis of 2D EM Scattering	p. 303
C.0 Summary	p. 303
C.1 Introduction of MoM	p. 303
C.2 Solution of a Two-Dimensional Scattering Problem	p. 305
C.2.1 Development of the Integral Equation and the MoM Solution	p. 305
C.2.2 Evaluation of the Parameter of Interest	p. 308
C.3 Implementation of a Serial MoM Code	p. 309
C.3.1 Flowchart and Results of a Serial MoM Code	p. 309
C.3.2 A Serial MoM Source Code for the 2D Scattering Problem	p. 312
C.4 Implementation of a Parallel MoM Code	p. 313
C.4.1 Flowchart and Results of a Parallel MoM Code	p. 313
C.4.2 A Parallel MoM Source Code Using ScaLAPACK for the 2D Scattering Problem	p. 318
C.5 Compilation and Execution of the Parallel Code	p. 331
C.6 Conclusion	p. 333
References	p. 333
Index	p. 335

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